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Periodic points of holomorphic maps via Lefschetz numbers

Authors: Núria Fagella and Jaume Llibre
Journal: Trans. Amer. Math. Soc. 352 (2000), 4711-4730
MSC (2000): Primary 55M20; Secondary 32H50, 37B99
Published electronically: June 8, 2000
MathSciNet review: 1707699
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Abstract: In this paper we study the set of periods of holomorphic maps on compact manifolds, using the periodic Lefschetz numbers introduced by Dold and Llibre, which can be computed from the homology class of the map. We show that these numbers contain information about the existence of periodic points of a given period; and, if we assume the map to be transversal, then they give us the exact number of such periodic orbits. We apply this result to the complex projective space of dimension $n$ and to some special type of Hopf surfaces, partially characterizing their set of periods. In the first case we also show that any holomorphic map of ${\mathbb CP}(n)$ of degree greater than one has infinitely many distinct periodic orbits, hence generalizing a theorem of Fornaess and Sibony. We then characterize the set of periods of a holomorphic map on the Riemann sphere, hence giving an alternative proof of Baker's theorem.

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Additional Information

Núria Fagella
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain
Address at time of publication: Departament de Matemàtica Aplicada i Anàlisi, Gran Via 585, 08007 Barcelona, Spain

Jaume Llibre
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain

Keywords: Set of periods, periodic points, holomorphic maps, Lefschetz fixed point theory
Received by editor(s): October 28, 1998
Published electronically: June 8, 2000
Additional Notes: Both authors are partially supported by DGICYT grant number PB96-1153
Article copyright: © Copyright 2000 American Mathematical Society

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